Some Asymptotic Properties of a Kernel Bispectum Estimate with Different Multitapers

Abstract

Assume X1,X2,…,XN are realizations of N observations from a real-valued discrete parameter third-order stationary process Xt,t=0±1,±2,…, with bispectrum fXXX(λ1,λ2) where “−π≤λ1,λ2≤π”. Based on the previous assumption, L different multitapered biperiodograms IXXX(mt)j(λ1,λ2);j=1,2,…,L on overlapped segments (Xt(j);1≤t<N) can be constructed. Further, the mean and variance of the average of these different multitapered biperiodograms can be expressed as asymptotic expressions. According to different bispectral windows/kernels (Wβ(j)(α1,α2), where “−π⩽α1,α2⩽π” andβ is the bandwidth) and IXXX(mt)j(λ1,λ2), the bispectrum fXXX(λ1,λ2) can be estimated. The asymptotic expressions of the first- and second-ordered moments as well as the integrated relative mean squared error (IMSE) of this estimate are derived. Finally, some estimation results based on numerically generated data from the selected process “DCGINAR(1)” are presented and discussed in detail.